For \(s\in [2,4)\) , \(\Phi ^{Q}_{m,n}\)  reduces to \(\Phi \) . Here, the non-uniform Berry–Esseen bounds in [1]}, [2]} allow for better bounds in view of the argument \(x\)  than t.clt.
Without the factor \((1+|x|)^{m}\) , this result has been derived in [3]}, but only for integer \({s}\) . For \(Z_n\) , non-uniform bound in the spirit of ours can be found in [4]}, for instance.
